Physics theoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)

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Local energy representation and thermodynamic conjugates

Physical scope

Local single-phase, positive-temperature charts of the adopted macroscopic entropy representation; no global inverse or statement at a phase boundary is asserted.

Statement

Let $c$ be a material constitution with $r$ named species, and let $O \subseteq D_{c}$ be an open dimensionless single-phase subset from Equilibrium state domains. Its physical SI chart is $Ω = Φ_{c} (O)$ under the diagonal unit map $Φ_{c}$ defined in Entropic temperature, pressure, and molar chemical potentials. Let $(U_{0}, V_{0}, n_{0, 1}, \dots, n_{0, r}) \in Ω$ , with $U$ in joules, $V > 0$ in cubic metres, and each $n_{j} > 0$ in moles (SI quantities and unit conventions). On $Ω$ , write $S_{c} (U, V, n)$ for the physical-coordinate representative $S_{c} \circ Φ_{c}^{- 1}$ ; its physical-coordinate partials include the unit factors as specified in Concave macroscopic fundamental entropy. Assume this representative is $C^{2}$ and $S_{c, U} (U_{0}, V_{0}, n_{0}) > 0$ . Then there are open neighborhoods of $(S_{c} (U_{0}, V_{0}, n_{0}), V_{0}, n_{0})$ and $(U_{0}, V_{0}, n_{0})$ on which the equation

$S_{c} (U, V, n) = s$

has a unique solution $U = U_{c} (s, V, n)$ of class $C^{2}$ . On this local chart,

$U_{c, s} = T_{c}, U_{c, V} = - p_{c}, U_{c, n_{j}} = μ_{c, j},$

with the intensive functions defined by Entropic temperature, pressure, and molar chemical potentials. This is a local representation only: it asserts neither global injectivity or domain coverage nor differentiability at a nonsmooth phase boundary.

Facts & Assumptions

Given: A fixed constitution $c$ , an open dimensionless single-phase subset $O \subseteq D_{c}$ , its physical SI chart $Ω = Φ_{c} (O)$ , a point $(U_{0}, V_{0}, n_{0}) \in Ω$ with $V_{0} > 0$ and positive mole amounts, and the adopted $C^{2}$ physical-coordinate entropy representative with $S_{c, U} (U_{0}, V_{0}, n_{0}) > 0$ . Use the stated SI units and center-of-mass rest-frame energy convention.

[A1]

Concave macroscopic fundamental entropy: On each designated open single-phase chart, the entropy representative is $C^{2}$ and its finite partial derivative $S_{c, U}$ at fixed $V, n$ is strictly positive. This is an adopted physical premise, not a proved physical law.

[F1]

Equilibrium state domains, Galilean inertial charts and the center-of-mass rest frame, and Entropic temperature, pressure, and molar chemical potentials: $D_{c}$ consists of dimensionless SI numerical coordinates, and $Φ_{c}$ maps them to the physical-coordinate chart $Ω$ ; $S_{c} (U, V, n)$ denotes the entropy composed with $Φ_{c}^{- 1}$ . The energy $U$ is evaluated in the center-of-mass rest frame. The state-domain definition does not assert that every ambient coordinate tuple is physically realizable.

[F2]

Entropic temperature, pressure, and molar chemical potentials: $T_{c} = 1 / S_{c, U}$ , $p_{c} = T_{c} S_{c, V}$ , and $μ_{c, j} = - T_{c} S_{c, n_{j}}$ , with the other physical coordinates held fixed in each partial derivative.

[F3]

The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ : For differentiable maps on open Euclidean domains, differentiation of a composition obeys the total-derivative chain rule.

[F4]

$C^{k}$ Euclidean maps and diffeomorphisms: A scalar map is $C^{2}$ when its coordinate partial derivatives through order two exist and are continuous.

[F5]

$C^{k}$ Euclidean maps are closed under componentwise algebra and composition: Finite sums and products and compositions of $C^{1}$ Euclidean maps are $C^{1}$ .

[L1]

Local graph form at a nonsingular derivative minor: The supplier's Statement says that a $C^{1}$ map on an open Euclidean set with a zero and an invertible derivative minor has a unique local $C^{1}$ graph representation, with derivative $D φ = - (D_{v} F)^{- 1} D_{u} F$ . In its scalar case, apply it to $G (U, s, V, n) = S_{c} (U, V, n) - s$ , placing $U$ in the dependent-coordinate block; $G_{U} = S_{c, U} > 0$ . The supplier currently has draft status and supplies a $C^{1}$ graph, not a $C^{2}$ one; the argument below derives the extra regularity from the adopted $C^{2}$ entropy premise.

Proof

1.1givenA1F1L1construct

Put $y = (V, n_{1}, \dots, n_{r})$ and choose an open interval $J$ around $s_{0} = S_{c} (U_{0}, y_{0})$ . On the open set where $(U, y) \in Ω$ and $s \in J$ , define $G (U, s, y) = S_{c} (U, y) - s$ . By [A1] and [F1], this map is $C^{1}$ , $G (U_{0}, s_{0}, y_{0}) = 0$ , and its $1 \times 1$ derivative minor in the $U$ coordinate equals $S_{c, U} (U_{0}, y_{0}) > 0$ . Since the total number of coordinates is $r + 3 > 1$ , the scalar graph case of [L1] applies after placing $U$ in the dependent block. Conditional on that draft supplier, it gives a unique local $C^{1}$ function $U = U_{c} (s, y)$ solving $S_{c} (U_{c} (s, y), y) = s$ .

2.1givenA1F3F4F5L1step 1.1algebra

Differentiate $S_{c} (U_{c} (s, y), y) = s$ in each coordinate of $(s, y)$ using [F3]. Writing $Ψ (s, y) = (U_{c} (s, y), y)$ , the resulting first-derivative identities are $U_{c, s} = 1 / (S_{c, U} \circ Ψ)$ and $U_{c, y_{j}} = - (S_{c, y_{j}} \circ Ψ) / (S_{c, U} \circ Ψ)$ . Each partial $S_{c, U}, S_{c, y_{j}}$ is $C^{1}$ by [A1],[F4], and $Ψ$ is $C^{1}$ by [L1]. Thus their compositions with $Ψ$ are $C^{1}$ by [F3],[F5]. For a positive scalar $z$ , $ρ (z) = 1 / z$ is $C^{1}$ : its difference quotient is $[ρ (z + h) - ρ (z)] / h = - 1 / (z (z + h)) \to - 1 / z^{2}$ , a continuous derivative on $z > 0$ . Since $S_{c, U} \circ Ψ > 0$ , the displayed expressions for every first partial of $U_{c}$ are $C^{1}$ by [F5]. Hence all second coordinate partials of $U_{c}$ exist and are continuous, so $U_{c}$ is $C^{2}$ by [F4].

3.1F2step 2.1algebra∎

By [F2], $T_{c} = 1 / S_{c, U}$ , $p_{c} = T_{c} S_{c, V}$ , and $μ_{c, j} = - T_{c} S_{c, n_{j}}$ . The derivative identities in step 2.1 therefore give $U_{c, s} = T_{c}$ , $U_{c, V} = - p_{c}$ , and $U_{c, n_{j}} = μ_{c, j}$ , with the other coordinates held fixed.

Remarks

The mathematical supplier Local graph form at a nonsingular derivative minor supplies the local $C^{1}$ graph interface. The argument above derives $C^{2}$ regularity from the entropy's $C^{2}$ hypothesis; it does not attribute a $C^{2}$ conclusion to that supplier.

Local energy representation and thermodynamic conjugates

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources